Universal Equivariant Line Bundles

Question

Complex line bundles are classified by maps into the universal bundle $\gamma\rightarrow \mathbb{C}P^\infty$. If I wanted to talk about $G$-equivariant line bundles over $X$, is there a corresponding universal equivariant line bundle?

Answer 1

Yes, and the construction is similar to the nonequivariant case. The difference is that you want to consider an "equivariant infinite complex projective space" instead of the usual infinite complex projective space of complex lines in $\mathbb{C}^\infty$.

Let $V$ be the direct sum of countably many copies of each irreducible complex representation of $G$, let $BU_G$ be the space of $1$-dimensional subspaces of $V$, and let $EU_G$ be the space of pairs $(\ell,v)$ where $\ell \in BU_G$ and $v \in \ell$ (here $BU_G$ and $EU_G$ have the obvious $G$-actions). Then $\pi:EU_G \longrightarrow BU_G$, $\pi(\ell,v)=\ell$ is a universal $G$-equivariant line bundle.

The construction of a universal complex $n$-plane bundle (work with the "equivariant infinite Grassmannian" of complex $n$-planes in $V$ instead) and a universal real $n$-plane bundle (take $V$ to consist of irreducible real representations in this case) for any positive integer $n$ follow analogously with the nonequivariant case.